More inequalities on numerical radii of sectorial matrices

Abstract

In this article, we refine some numerical radius inequalities of sectorial matrices recently obtained by Bedrani , Kittaneh and Sababheh. Among other results, it is shown that if $A_i\in\mathbb{M}_n(\mathbb{C})$ with $W(A_i)\subseteq S_{\alpha}$, $i=1,2\cdots,n$, and $a_1,\cdots,a_n$ are positive real numbers with $\sum_{j=1}^na_j=1$ , then \begin{eqnarray*} \omega^t\left(\sum_{i=1}^n a_iA_i\right)\le\cos^{2t}(\alpha)\omega\left(\sum_{i=1}^n a_iA_i^t\right), \end{eqnarray*} where $t\in[-1,0]$. An improvement of the Heinz-type inequality for the numerical radii of sectorial matrices is also given. Moreover, we present some numerical radius inequalities of sectorial matrices involving positive linear maps.

Keywords

• sectorial matrices
• numerical radius
• function
• operator mean
• positive linear maps